## Friday, July 31, 2009

### Simple Newton's Method Fractal code in MATLAB

Due to popular request I've sharing some very simple Newton's Method Fractal code in MATLAB. It produces the following 800x800 image (in about 2.5 seconds on my 2.4Ghz Macbook Pro):

>> [niters,solutions] = matlab_fractal;
>> imagesc(niters)

function [niters,solutions] = matlab_fractal
%Create Newton's Method Fractal Image
%Tomasz Malisiewcz (tomasz@cmu.edu)
%http://quantombone.blogspot.com/
NITER = 40;
threshold = .001;

[xs,ys] = meshgrid(linspace(-1,1,800), linspace(-1,1,800));
solutions = xs(:) + i*ys(:);
select = 1:numel(xs);
niters = NITER*ones(numel(xs), 1);

for iteration = 1:NITER
oldi = solutions(select);

%in newton's method we have z_{i+1} = z_i - f(z_i) / f'(z_i)
solutions(select) = oldi - f(oldi) ./ fprime(oldi);

%check for convergence or NaN (division by zero)
differ = (oldi - solutions(select));
converged = abs(differ) < threshold;
problematic = isnan(differ);

niters(select(converged)) = iteration;
niters(select(problematic)) = NITER+1;
select(converged | problematic) = [];
end

niters = reshape(niters,size(xs));
solutions = reshape(solutions,size(xs));

function res = f(x)
res = (x.^2).*x - 1;

function res = fprime(x)
res = 3*x.^2;

1. Just so you know: This code runs in Octave as well.

2. I imagined that it would run in Octave too but never tried. Thanks for trying out!

The reason why it runs fairly fast is that the Newton's Method iterations are vectorized over the pixels in the image. The only loop is over iterations. My additional speedup is to remove pixels from consideration once they have converged.

What's happening is that a complex number gets set for every pixel in the image and Newton's Method is ran to solve for the roots of the equation z^3-1=0 with the pixel's complex number as the initialization.

This image is then created by showing for every pixel how many iterations it took to converge. The other quantity of interest is which root was found; z^3-1=0 has three roots, one real and two complex. Showing the roots is a bit harder and to keep the code short and readable I used the iterations and Matlab's imagesc jet colorscheme.

3. hello sir, can you give me the MATLAB code for finding the complex (imaginary roots ) of a function? e.g --- x^2+25=0.

4. You can easily adapt the above code to do that for you via Newton's method. Just define f(x) and fprime(x) which is the derivative of f(x). I can't help you with the details -- it is not difficult.

5. If you type "imagesc(niters)" you will generate the picture

6. Hi. What is the licence of your code ?

Copyright (C) 2011 by Tomasz Malisiewicz

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

8. can u plzz send me a simple fractal program in matlab?